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User blog:B1mb0w/The Quantum Function
'The Quantum Function' My new Quantum function is the fastest growing function I have defined. The Quantum function is a set of two functions \(Q()\) and \(t()\) and has a growth rate well beyond \(f_{\vartheta(\Omega\uparrow\uparrow\omega)}(n)\) 'Notation Explained' I use notation that is not in general use. For example parameter subscript brackets, leading zeros assumption, recursion parameter subscript \(*\), and the decremented function \(C\). Refer to this blog for a complete explanation of this notation. 'Defining the \(Q\) function' Using my notation to define the \(Q\) function. \(Q© = c + 1\) \(Q(a_{x},c + 1,n) = Q^C(a_{x},c,n_*)\) \(Q(1,0_{+ 1},n) = Q(t_C(0),0_{c},n)\) \(Q(a_{x},c + 1,0_{+ 1},n) = Q(a_{x},c,t_C(0),0_{z},n)\) Note that \(C\) is defined in my blog on notation. 'Defining the \(t\) function' Using my notation to define the \(t\) function. \(t_0(0) = n\) The \(t\) function is used to substitute for values of \(n\) in any given Quantum function in the form: \(Q(a_{x},n)\) then \(t_0(0) = n\) More precise examples are: \(Q(1,t_0(0),n) = Q(1,n,n)\) \(Q^2(t_0(0),n_*) = Q(t_0(0),Q(t_0(0),n)) = Q(n,Q(t_0(0),n))\) The definition then continues: \(t_0(c + 1) = Q^C(C_*,C)\) \(t_d(1,0_{+ 1},c + 1) = t_d^C(1_*,0_{y},c)\) \(t_d(a + 1,0_{+ 1},c + 1) = t_d^C(a,0_*,0_{y},C)\) \(t_d(a_{x},b + 1,c + 1) = t_d^C(a_{x},b,C_*)\) \(t_d(1,0_{+ 1}) = t_d^C(1_*,0_{c})\) \(t_d(a_{x},c + 1,0_{+ 1}) = t_d^C(a_{x},c,0_*,0_{z})\) \(t_{c + 1}(0) = t_c(1,0_{C})\) \(t_{d + 1}(c + 1) = t_d(1,0_{C})\) 'Some Example \(Q\) Function Calculations' The Quantum function is similar to the fast growing hierarchy up to a point. \(Q(3) = 4\) \(Q(1,n) = Q^{Q(0,n)}(0,n_*) = Q^{Q(n)}(n) = n + n + 1 = n.2 + 1 > f_1(n)\) \(Q(2,n) > f_2(n)\) In general: \(Q^h(g,n_*) > f_g^h(n)\) The \(t\) function and in particular the \(t_0(0)\) function acts like a wildcard which allows substitution, similar to the role of ordinals, like \(\omega\) and the Veblen function in the fast growing hierarchy. Here is the simplest example. \(Q(t_0(0),n) > f_{\omega}(n)\) \(Q(Q(3,t_0(0)),n) > f_{\varphi(1,0)}(n)\) \(Q(Q(t_0(0),t_0(0)),n) \approx f_{\varphi(\omega,0)}(n)\) \(Q(Q^2(1_*,t_0(0)),n) = Q(Q(Q(1,t_0(0)),t_0(0)),n) > f_{\varphi(\omega.2,0)}(n)\) \(Q(Q^2(2_*,t_0(0)),n) > f_{\varphi(\omega^2,0)}(n)\) \(Q(Q^2(3_*,t_0(0)),n) > f_{\varphi(\varphi(1,0),0)}(n) > f_{\varphi^2(1_*,0)}(n)\) \(Q(Q^3(3_*,t_0(0)),n) > f_{\varphi^3(1_*,0)}(n)\) \(Q(t_0(1),n) > Q(Q^{Q(t_0(0))}(3_*,t_0(0)),n) > f_{\varphi^{\omega}(1_*,0)}(n) = f_{\varphi(1,0,0)}(n)\) \(Q(Q(t_0(0),t_0(1)),n) > f_{\varphi(\omega,\varphi(1,0,0) + 1)}(n)\) \(Q(Q(t_0(1),t_0(1)),n) > f_{\varphi(1,0,1)}(n)\) \(Q(Q^{t_0(0)}(t_0(1),t_0(1)_*),n) > f_{\varphi(1,0,\omega)}(n)\) \(Q(Q(Q(t_0(1)),t_0(1)),n) = Q(Q^{Q(t_0(1))}(t_0(1),t_0(1)_*),n) > f_{\varphi^2(1,0,0_*)}(n)\) \(Q(Q^2(Q(t_0(1)),t_0(1)_*),n) > f_{\varphi^3(1,0,0_*)}(n)\) \(Q(Q^{t_0(0)}(Q(t_0(1)),t_0(1)_*),n) > f_{\varphi(1,1,0)}(n)\) \(Q(Q^{Q(1,t_0(0))}(Q(t_0(1)),t_0(1)_*),n) > f_{\varphi(1,1,1)}(n)\) \(Q(Q(Q^2(t_0(1)),t_0(1)),n) = Q(Q^{t_0(1)}(Q(t_0(1)),t_0(1)_*),n) > f_{\varphi(1,1,\varphi(1,0,0))}(n)\) \(Q(Q(Q^3(t_0(1)),t_0(1)_*),n) > Q(Q^{t_0(0)}(Q^2(t_0(1)),t_0(1)_*),n) > f_{\varphi(1,2,0)}(n)\) \(Q(Q(Q(1,t_0(1)),t_0(1)),n) = Q(Q(Q^{Q(t_0(1))}(t_0(1)),t_0(1)_*),n) > f_{\varphi^2(1,0_*,0)}(n)\) \(Q(Q^3(1_*,t_0(1)),n) = Q(Q(Q(Q(1,t_0(1))),t_0(1)),n) > f_{\varphi^3(1,0_*,0)}(n)\) \(Q(t_0(2),n) > Q(Q^{Q(t_0(0))}(1_*,t_0(1)),n) > f_{\varphi(2,0,0)}(n)\) \(Q(t_0(m),n) > f_{\varphi(m,0,0)}(n)\) \(Q(t_0^2(1),n) = Q(t_0(t_0(1)),n) > f_{\varphi^2(1_*,0,0)}(n)\) \(Q(t_0(1,0),n) > f_{\varphi(1,0,0,0)}(n)\) \(Q(t_1(0),n) = Q(t_0(1,0_{t_0(0)}),n) > f_{svo}(n)\) '\(Q\) Function Calculations beyond svo' \(Q(t_1(0),n) > f_{svo}(n) = f_{\vartheta(\Omega^{\vartheta(1)})}(n)\) \(Q(t_1(1),n) = Q(t_0(1,0_{t_1(0)}),n) > f_{\vartheta(\Omega^{\vartheta(\Omega^{\vartheta(1)})})}(n)\) \(Q(t_1(t_0(0)),n) > f_{\vartheta(\Omega^{\Omega})}(n)\) \(Q(t_1^2(0),n) > Q(t_1(t_0(0)),n) > f_{\vartheta(\Omega^{\Omega})}(n)\) \(Q(t_1(1,0),n) = Q(t_1^{t_1(1)}(1),n) > f_{\vartheta(\Omega\uparrow\uparrow\vartheta(1))}(n)\) \(Q(t_2(0),n) = Q(t_1(1,0_{t_1(0)}),n) > f_{\vartheta(\Omega\uparrow\uparrow\vartheta(1))}(n)\) \(Q(1,0,n) = Q(t_{Q(1,n)}(0),n) > f_{\vartheta(\Omega\uparrow\uparrow\vartheta(1))}(n)\) 'Simple \(Q\) Function Calculations where \(n = 1\)' \(Q(1) = 2\) \(Q(t_0(0),1) = Q(1,1) = 3\) \(Q(Q(t_0(0)),1) = Q^{Q(t_0(0),1)}(t_0(0),1_*) = Q^3(t_0(0),1_*) = Q^2(t_0(0),3_*) > f_{\omega}^2(3)\) \(Q(t_0(1),1) = Q(Q(t_0(0),t_0(0)),1) = Q(Q(1,t_0(0)),1) = Q(Q^{Q(t_0(0))}(t_0(0)),1)\) \(=Q^{Q(1)}(Q(t_0(0)),1_*) = Q^2(Q(t_0(0)),1_*) > Q(Q(t_0(0)),f_{\omega}^2(3)) > f_{\omega + 1}(f_{\omega}^2(3))\) \(> f_{\omega + 1}(googol) > f_{\omega + 1}(3)\) \(Q(Q(t_0(1)),1) = Q^{Q(1)}(t_0(1),1_*) = Q^2(t_0(1),1_*) > Q(t_0(1),3) > f_{\varphi(1,0,0)}(3)\) \(Q(t_0(1,0),1) = Q(t_0^{t_0(1)}(t_0(1)),1) = Q(t_0^{Q^{Q(t_0(0))}(t_0(0))}(t_0(1)),1)\) \(= Q(t_0^{Q^2(t_0(0))}(t_0(1)),1) > Q(t_0^{Q(t_0(0))}(t_0(1)),3) > Q(t_0^4(t_0(1)),3)\) \(> f_{\varphi^4(1_*,0,0)}(3) > f_{\varphi(1,0,0,0)}(3) > f_{svo}(3)\) 'Calculating \(Q(1,0,1)\)' The Quantum function is similar to the fast growing hierarchy up to a point. But both the \(Q\) and \(t\) functions grow faster than the comparable Fast Growing hierarchy and Veblen functions. This is clear in the previous examples, where even if \(n = 1\) the Quantum function will still grow quickly. The fast growing hierarchy doesn't grow at all when \(n = 1\) \(Q(Q(t_0(0)),1) > f_{\omega}^2(3)\) \(f_{\omega + 1}(1) = f^1_{\omega}(1) = f_{1}(1) = f^1_{0}(1) = f_{0}(1) = 2\) Calculating \(Q(1,0,1)\) illustrates how fast the Quantum Function can grow. \(Q(1,0,1) = Q(t_{Q(1,1)}(0),1) = Q(t_3(0),1) = Q(t_2(1,0_{t_2(0)}),1)\) \(= Q(t_2(1,0_{[t_1(1,0_{[t_0(1,0_{t_0(0)})]})]}),1)\) \(= Q(t_2(1,0_{[t_1(1,0_{[t_0(1,0_{1})]})]}),1)\) \(= Q(t_2(1,0_{[t_1(1,0_{t_0(1,0)})]}),1)\) 'Calculating \(Q(2,0,1)\)' \(Q(2,0,1) = Q(1,t_{Q(1,0,1)}(0),1) = Q(1,t_{Q(t_3(0),1)}(0),1)\) 'Further References' Further references to relevant blogs can be found here: User:B1mb0w Category:Blog posts